A_p Weights and quantitative estimates in the Schrödinger setting

Suppose L= - Δ + V is a Schrödinger operator on Rⁿ with a potential V belonging to certain reverse Hölder class RH_σ with σ≥ n/ 2. The aim of this paper is to study the A_p weights associated to L, denoted by ApL, which is a larger class than the classical Muckenhoupt A_p weights. We first prove the quantitative ApL bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative Ap,qL bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical A_{p , q} constant. However, since Ap,q⊂Ap,qL, the Ap,qL constants are smaller than A_{p , q} constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the “exp–log” link between ApL and BMO_L (the BMO space associated with L), and show that for w∈ApL, log w is in BMO_L, and that the reverse is not true in general.